The Project Gutenberg EBook of Solid Geometry with Problems and Applications (Revised edition), by H. E. Slaught
and N. J. Lennes This
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Solid Geometry with Problems and Applications (Revised edition) Author: H. E. Slaught N. J. Lennes Release Date: August 26, 2009 [EBook #29807] Language: English Character set encoding: ISO-8859-1 *** START
OF THIS
PROJECT GUTENBERG EBOOK SOLID GEOMETRY *** Bonaventura Cavalieri (1598–1647) was one
of the most inﬂuential mathematicians
of his time. He was chieﬂy noted for his invention
of the so-called “Principle
of Indivisibles” by which he derived areas
and volumes. See pages 143
and 214.
SOLID GEOMETRY WITH PROBLEMS AND APPLICATIONS REVISED EDITION BY H. E. SLAUGHT, Ph.D., Sc.D. PROFESSOR
OF MATHEMATICS IN
THE UNIVERSITY
OF CHICAGO
AND N. J. LENNES, Ph.D. PROFESSOR
OF MATHEMATICS IN
THE UNIVERSITY
OF MONTANA ALLYN
and BACON Bo<on New York Chicago Produced by Peter Vachuska, Andrew D. Hwang, Chuck Greif
and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber’s note
The original book is copyright, 1919, by H. E. Slaught
and N. J. Lennes. Figures may have been moved
with respect to
the surrounding text. Minor typographical corrections
and presentational changes have been made without comment. This PDF ﬁle is formatted for printing, but may be easily recompiled for screen viewing. Please see
the preamble
of the L A T E X source ﬁle for instructions. PREFACE In re-writing
the Solid Geometry the authors have consistently car- ried out
the distinctive features described in
the preface
of the Plane Geometry. Mention is here made only
of certain matters which are particularly emphasized in
the Solid Geometry. Owing to
the greater maturity
of the pupils it has been possible to make
the logical structure
of the Solid Geometry more prominent than in
the Plane Geometry.
The axioms are stated
and applied at
the precise points where they are to be used. Theorems are no longer quoted in
the proofs but are only referred to by paragraph numbers; while
with increasing frequency
the student is left to his own devices in supplying
the reasons
and even in ﬁlling in
the logical steps
of the argument. For convenience
of reference
the axioms
and theorems
of plane
geometry which are used in
the Solid Geometry are collected in
the Introduction. In order to put
the essential principles
of solid geometry, together
with a reasonable number
of applications, within limited bounds (156 pages), certain topics have been placed in an Appendix. This was done in order to provide a minimum course in convenient form for class use
and not because these topics, Similarity
of Solids
and Applications of Projection, are regarded as
of minor importance. In fact, some
of the examples under these topics are among
the most interesting
and concrete in
the text. For example, see pages 180–183, 187–188, 194– 195.
The exercises in
the main body
of the text are carefully graded as to diﬃculty
and are not too numerous to be easily performed.
The concepts
of three-dimensional space are made clear
and vivid by many simple illustrations
and questions under
the suggestive headings “Sight PREFACE Work.” This plan
of giving many
and varied simple exercises, so eﬀec- tive in
the Plane Geometry, is still more valuable in
the Solid Geometry where
the visualizing
of space relations is diﬃcult for many pupils.
The treatment
of incommensurables throughout
the body
of this text, both Plane
and Solid, is believed to be sane
and sensible. In each case, a frank assumption is made as to
the existence
of the concept in question (length
of a curve, area
of a surface, volume
of a solid)
and of its realization for all practical purposes by
the approximation process. Then, for theoretical completeness, rigorous proofs
of these theorems are given in Appendix III, where
the theory
of limits is presented in far simpler terminology than is found in current text-books
and in such a way as to leave nothing to be unlearned or compromised in later mathematical work. Acknowledgment is due to Professor David Eugene Smith for
the use
of portraits from his collection
of portraits
of famous mathematicians. H. E. SLAUGHT N. J. LENNES Chicago
and Missoula, May, 1919. CONTENTS INTRODUCTION 1 Space Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Axioms
and Theorems from Plane
Geometry . . . . . . . . . . 5 BOOK I. Properties
of the Plane 10 Perpendicular Planes
and Lines . . . . . . . . . . . . . . . . . 11 Parallel Planes
and Lines . . . . . . . . . . . . . . . . . . . . . 21 Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Constructions
of Planes
and Lines . . . . . . . . . . . . . . . . 37 Polyhedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . 42 BOOK II. Regular Polyhedrons 53 Construction
of Regular Polyhedrons . . . . . . . . . . . . . . 56 BOOK III. Prisms
and Cylinders 58 Properties
of Prisms . . . . . . . . . . . . . . . . . . . . . . . 59 Properties
of Cylinders . . . . . . . . . . . . . . . . . . . . . . 75 BOOK IV. Pyramids
and Cones 85 Properties
of Pyramids . . . . . . . . . . . . . . . . . . . . . . 86 Properties
of Cones . . . . . . . . . . . . . . . . . . . . . . . . 98 BOOK V.
The Sphere 113 Spherical Angles
and Triangles . . . . . . . . . . . . . . . . . 125 Area
of the Sphere . . . . . . . . . . . . . . . . . . . . . . . . 143 Volume
of the Sphere . . . . . . . . . . . . . . . . . . . . . . . 150 APPENDIX TO
SOLID GEOMETRY I. Similar Solids . . . . . . . . . . . . . . . . . . . . . . . . . 168 II.
Applications of Projection . . . . . . . . . . . . . . . . . . 183 III. Theory
of Limits . . . . . . . . . . . . . . . . . . . . . . . . 196 INDEX 217 PORTRAITS
AND BIOGRAPHICAL SKETCHES Cavalieri . . . . . . . . . . . . . . . . . . . . . . . . . Frontispiece Thales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Legendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
SOLID GEOMETRY [...]... that
the lower
and right-hand sides are nearer
the observer than
the other edges Hence,
the plane represented does not lie in
the plane
of the paper, but
the lower part
of it stands out toward
the observer
The ﬁgure ABCD represents a triangular pyramid
The corner marked B is nearest
the observer
and this is indicated by
the heavy lines
The triangle ACD lies behind
the pyramid
and is thus farther from the. .. equal if
the hypotenuse
and a side
of one are equal respectively to
the hypotenuse
and a side
of the other 41 Two right triangles are equal if a side
and an acute angle
of one are equal respectively to
the corresponding side
and acute angle
of the other 42 Two right triangles are equal if
the hypotenuse
and an acute angle
of one are equal respectively to
the hypotenuse
and an acute angle
of the other 43... area
of a parallelogram is equal to
the product
of its base
and altitude 59 Two parallelograms have equal areas if they have equal bases
and equal altitudes 60
The area
of a triangle is equal to one half
the product
of its base
and altitude 61 If a is a side
of a triangle
and h
the altitude on it
and b another side
and k
the altitude on it, then ah = bk 62
The area
of a trapezoid is equal to one half the. ..
of the segment AB 5 Find
the locus
of all points equidistant from two given points A
and B,
and also equidistant from two points C
and D Discuss 6 State
and prove a theorem
of solid geometry corresponding
to the theorem
of plane
geometry given in § 38 7 If in
the ﬁgure P D ⊥ plane M ,
and DC ⊥ AB, a line
of the plane M , prove that P C ⊥ AB Suggestion Lay oﬀ CA = CB,
and compare triangles 8 If in the. .. outside
of each
of two non-parallel lines there is one
and only one plane parallel to both
of these lines 100 Theorem XIII PROPERTIES
OF THE PLANE 25 Given a point P outside
of the non-parallel lines l1
and l2 To prove that there is one
and only one plane M through P parallel to l1
and l2 Proof : Through P pass l3 l1
and l4 l2 Then
the plane M
of l3
and l4 is parallel to l1
and l2 In any other plane... in plane or in
solid geometry? 7 Find
the locus
of all points equidistant from two given parallel planes Is this a problem in plane or in
solid geometry? 8
The side walls
of your schoolroom meet each other in four vertical lines Are any two
of these parallel? Are any three
of them parallel? Do any three
of them lie in
the same plane? 9
The side walls
of your schoolroom meet
the ﬂoor
and the ceiling in... bisects
the other side also 2 Show that a plane containing one only
of two parallel lines is parallel to
the other 3 If in two intersecting planes a line
of one is parallel to a line
of the other, then each
of these lines is parallel to
the line
of intersection
of the planes 4 Show that three lines which do not meet in one point must all lie in
the same plane if each intersects
the other two 26
SOLID GEOMETRY: ... angles 31
The sum
of all consecutive angles about a point
and on one side
of a straight line is two right angles 32 If two adjacent angles are supplementary, their exterior sides lie in
the same straight line 33 If in two triangles two sides
of one are equal respectively to two sides
of the other, but
the third side
of the ﬁrst is greater than
the third side
of the second, then
the included angle
of the. .. parallel lines l1
and l2 To prove that in each case a plane is determined Proof : (1) Let A
and B be two points on l Then one
and only one plane M can be passed through l
and P because one
and only one plane can be passed through A, B,
and P Ax 2, § 66 (2) Let A be
the intersection point
of l1
and l2 ,
and B
and C any other points, one on l1
and the other on l2 Then A, B,
and C determine
the plane N in... to
the plane
of these lines 76 Theorem III Given a line l perpendicular to each
of the lines l1
and l2 at
the point P To prove that l is perpendicular to
the plane
of l1
and l2 Proof : Let M be
the plane
of l1
and l2 ,
and let l3 be any line in M through P Draw a line meeting l1 , l2 ,
and l3 in
the points B, C,
and D respectively Let E
and F be points on l, on opposite sides
of P ,
and such that . The Project Gutenberg EBook of Solid Geometry with Problems and Applications (Revised edition), by H. E. Slaught and N. J. Lennes This eBook is for the use of anyone anywhere at no cost and with almost. or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www .gutenberg. org Title: Solid Geometry with Problems and Applications (Revised edition) Author:. emphasized in the Solid Geometry. Owing to the greater maturity of the pupils it has been possible to make the logical structure of the Solid Geometry more prominent than in the Plane Geometry. The axioms